Part 2

Question 6

Find the compound interest and compound amount of Rs. 2,00,000 invested for 3 years such that the rate of interest for the first year is 8% p.a., for the second year it is 10% p.a. and for the third year it is 12%.

Solution:

Given information:

• Principal $(P) = \text{Rs. } 2,00,000$
• Time $(T) = 3$ years
• Rate of interest for the first year $(R_1) = 8\%$
• Rate of interest for the second year $(R_2) = 10\%$
• Rate of interest for the third year $(R_3) = 12\%$

We need to find:

• Compound amount $(CA) = ?$
• Compound interest $(CI) = ?$

Finding the Compound Amount:

According to the formula, compound amount when rates are different for different years:

$CA = P\left(1 + \frac{R_1}{100}\right)\left(1 + \frac{R_2}{100}\right)\left(1 + \frac{R_3}{100}\right)$

Substitute the given values:

$CA = 2,00,000\left(1 + \frac{8}{100}\right)\left(1 + \frac{10}{100}\right)\left(1 + \frac{12}{100}\right)$

Simplify step by step:

$CA = 2,00,000\left(\frac{108}{100}\right)\left(\frac{110}{100}\right)\left(\frac{112}{100}\right)$
$CA = 2,00,000(1.08)(1.10)(1.12)$
$CA = 2,00,000 \times 1.33056$
$CA = \text{Rs. } 2,66,112$

Finding the Compound Interest:

Thus, compound interest $(CI) = CA - P$

$CI = \text{Rs. } 2,66,112 - \text{Rs. } 2,00,000 = \text{Rs. } 66,112$

Step-by-step verification:

Let's verify by calculating year by year:

• End of 1st year: $A_1 = 2,00,000 \times 1.08 = \text{Rs. } 2,16,000$
• End of 2nd year: $A_2 = 2,16,000 \times 1.10 = \text{Rs. } 2,37,600$
• End of 3rd year: $A_3 = 2,37,600 \times 1.12 = \text{Rs. } 2,66,112$ ✓

Final Answers:

• Compound Amount = Rs. 2,66,112
• Compound Interest = Rs. 66,112

Note: This problem demonstrates compound interest with variable rates, which is more realistic in real-world scenarios where interest rates may change over time.

Question 7

A sum amounts to Rs. 14,520 in 2 years and Rs. 15,972 in 3 years at a certain rate of annual compound interest. Then,

1. Find the rate of compound interest.
2. Find what is the principal.

Solution:

Given information:

• Rate of interest $(R) = R\%$ (unknown)
• Principal $(P) = \text{Rs. } x$ (unknown)

Case I:

• Compound amount $(CA_1) = \text{Rs. } 14,520$
• Time $(T) = 2$ years

Using the formula, compound amount $(CA_1) = P\left(1 + \frac{R}{100}\right)^T$

$\text{Rs. } 14,520 = x\left(1 + \frac{R}{100}\right)^2$ .................(i)

Case II:

• Compound amount $(CA_2) = \text{Rs. } 15,972$
• Time $(T) = 3$ years

Using the formula, compound amount $(CA_2) = P\left(1 + \frac{R}{100}\right)^T$

$\text{Rs. } 15,972 = x\left(1 + \frac{R}{100}\right)^3$ .................(ii)

Finding the rate of compound interest:

Dividing equation (ii) by (i), we get

$\frac{15,972}{14,520} = \frac{x\left(1 + \frac{R}{100}\right)^3}{x\left(1 + \frac{R}{100}\right)^2}$
$1.10 = 1 + \frac{R}{100}$
$1.10 - 1 = \frac{R}{100}$
$0.10 \times 100 = R$ $R = 10$

∴ Rate of interest (R) = 10% p.a.

Finding the principal:

Again, putting $(R) = 10\%$ in equation (i), we get

$14,520 = x\left(1 + \frac{10}{100}\right)^2$
$14,520 = x(1.10)^2$
$14,520 = x \times 1.21$
$x = \frac{14,520}{1.21} = 12,000$

∴ Principal (P) = x = Rs. 12,000

Final Answers:

• (a) Rate of compound interest = 10% p.a.
• (b) Principal = Rs. 12,000

Verification:

• After 2 years: $12,000 \times (1.10)^2 = 12,000 \times 1.21 = \text{Rs. } 14,520$ ✓
• After 3 years: $12,000 \times (1.10)^3 = 12,000 \times 1.331 = \text{Rs. } 15,972$ ✓

Key Insight: This problem demonstrates how to solve for multiple unknowns by using the relationship between compound amounts at different time periods. The ratio of consecutive compound amounts gives us $(1 + R/100)$.

Question 8

A person deposited Rs. 2,00,000 in a development bank for 2 years to get the half yearly compound interest at the rate of 10% per annum after deducting the 5% tax on the interest. But right after a year, bank has changed the policy and decided to accomplish the interest terminally at the same rate of interest.

1. Find the interest of the first year by deducting the tax.
2. What would be the interest of the second year after deducting the tax?
3. What is the difference between interests of the first year and second year after deducting the tax? Find.
4. After deducting the tax, by what percentage the interest of the first year differ from the interest of the second year?

Solution:

Given information:

• Principal $(P) = \text{Rs. } 2,00,000$
• Rate of interest $(R) = 10\%$ p.a.
• Tax on interest = 5%
• First year: Half yearly compounding
• Second year: Quarterly compounding (after policy change)

(a) For the first year, the half yearly compound interest:

Using the half yearly compound interest formula:

$CI_1 = P\left[\left(1 + \frac{R}{200}\right)^{2T} - 1\right]$

Substitute the values:

$CI_1 = 2,00,000\left[\left(1 + \frac{10}{200}\right)^{2 \times 1} - 1\right]$
$CI_1 = 2,00,000\left[\left(\frac{210}{200}\right)^{2 \times 1} - 1\right]$
$CI_1 = 2,00,000\left[(1.05)^2 - 1\right]$
$CI_1 = 2,00,000[1.1025 - 1]$
$CI_1 = 2,00,000 \times 0.1025 = \text{Rs. } 20,500$

After deducting 5% tax:

$CI_1 = \text{Rs. } 20,500 - \text{Rs. } 20,500 \times \frac{5}{100}$
$CI_1 = \text{Rs. } 20,500 - \text{Rs. } 1,025 = \text{Rs. } 19,475$

∴ Interest of the first year after deducting tax is Rs. 19,475

(b) For the second year (quarterly compounding):

Compound amount after first year = Rs. 2,00,000 + Rs. 19,475 = Rs. 2,19,475

Now, principal for the second year = compound amount of the first year = Rs. 2,19,475

According to the quarterly compound interest formula:

$CI_2 = P\left[\left(1 + \frac{R}{400}\right)^{4T} - 1\right]$

Substitute the values:

$CI_2 = 2,19,475\left[\left(1 + \frac{10}{400}\right)^{4 \times 1} - 1\right]$
$CI_2 = 2,19,475\left[(1.025)^4 - 1\right]$
$CI_2 = 2,19,475[1.103812890625 - 1]$
$CI_2 = 2,19,475 \times 0.103812890625$
$CI_2 = \text{Rs. } 22,784.33$

Again, the interest after deducting 5% tax:

$CI_2 = 22,784.33 - 22,784.33 \times \frac{5}{100}$
$CI_2 = 22,784.33 - 1,139.21$
$CI_2 = \text{Rs. } 21,645.12$

∴ The interest after deducting 5% tax is Rs. 21,645.12

(c) The difference in interests:

Difference = $CI_2 - CI_1 = 21,645.12 - 19,475 = \text{Rs. } 2,170.12$

(d) The difference of interests in percentage:

Percentage difference = $\frac{CI_2 - CI_1}{CI_1} \times 100\%$

$= \frac{2,170.12}{19,475} \times 100$

Thus, the interest of the second year differs by 11.14% than that of the first year.

Final Answers:

• (a) First year interest after tax = Rs. 19,475
• (b) Second year interest after tax = Rs. 21,645.12
• (c) Difference in interests = Rs. 2,170.12
• (d) Percentage difference = 11.14%

Key Learning: This problem shows how different compounding frequencies (half-yearly vs quarterly) affect the final interest amount, and how tax deductions impact the actual returns on investments.

Question 9

A commercial bank releases a loan of Rs. 52,500 to Babulal and Jibanlal at the rate of yearly 10% compound interest. If the compound amount paid by Babulal in 2 years is the same as the compound amount paid by Jibanlal in 3 years, how much loan did each of them borrow from the bank?

Solution:

Given information:

• Total loan amount = Rs. 52,500
• Rate of interest = 10% p.a. (compound)
• Babulal's compound amount in 2 years = Jibanlal's compound amount in 3 years

Let:

• The loan amount of Babulal $(P_1) = \text{Rs. } x$
• The loan amount of Jibanlal $(P_2) = \text{Rs. } (52,500 - x)$

Compound amount to be paid by Babulal in 2 years:

$CA_1 = P_1\left(1 + \frac{R}{100}\right)^T$
$CA_1 = x\left(1 + \frac{10}{100}\right)^2$
$CA_1 = x\left(\frac{110}{100}\right)^2$
$CA_1 = x(1.10)^2 = 1.21x$

Compound amount to be paid by Jibanlal in 3 years:

$CA_2 = P_2\left(1 + \frac{R}{100}\right)^T$
$CA_2 = (52,500 - x)\left(1 + \frac{10}{100}\right)^3$
$CA_2 = (52,500 - x)\left(\frac{110}{100}\right)^3$
$CA_2 = (52,500 - x)(1.10)^3$
$CA_2 = (52,500 - x) \times 1.331$
$CA_2 = 69,877.5 - 1.331x$

Finding the individual loan amounts:

Now, according to the question, $CA_1 = CA_2$

$1.21x = 69,877.5 - 1.331x$
$1.21x + 1.331x = 69,877.5$
$2.541x = 69,877.5$
$x = \frac{69,877.5}{2.541} = 27,500$

Therefore:

• Babulal borrowed Rs. 27,500
• Jibanlal borrowed Rs. (52,500 - 27,500) = Rs. 25,000

Verification:

• Babulal's compound amount in 2 years: $27,500 \times (1.10)^2 = 27,500 \times 1.21 = \text{Rs. } 33,275$
• Jibanlal's compound amount in 3 years: $25,000 \times (1.10)^3 = 25,000 \times 1.331 = \text{Rs. } 33,275$ ✓

Final Answer:

Babulal borrowed Rs. 27,500 and Jibanlal borrowed Rs. 25,000 from the bank.

Key Insight: This problem demonstrates how different time periods with the same interest rate can result in equal compound amounts when the principal amounts are appropriately distributed.